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Given a finite presentation for a group G which can also be realised as a relative presentation we give conditions on the relative presentation which, if satisfied, proves G hyperbolic. Using a curvature distribution method we confirm these conditions for the length four one-relator relative presentation 〈 u , t | t n , t m u t u − r 〉 for many values of r and n deduce that the corresponding generalised Fibonacci groups F (r , n) are hyperbolic. [ABSTRACT FROM AUTHOR]

This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman ['Word hyperbolic semigroups', Math. Proc. Cambridge Philos. Soc. 136 (3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a $\mathbf {C}$ -multiplication table, where $\mathbf {C}$ is any reversal-closed super- $\operatorname {\mathrm {AFL}}$. In particular, we deduce that the free product of two groups with $\mathbf {ET0L}$ with respect to indexed multiplication tables again has an $\mathbf {ET0L}$ with respect to an indexed multiplication table. [ABSTRACT FROM AUTHOR]

The main objects of study in this article are pairs (G , H) where G is a topological group with a compact open subgroup, and H is a finite collection of open subgroups. We develop geometric techniques to study the notions of G being compactly generated and compactly presented relative to H. This includes topological characterizations in terms of discrete actions of G on complexes, quasi-isometry invariance of certain graphs associated to the pair (G , H) when G is compactly generated relative to H , and extensions of known results from the discrete case. For example, generalizing results of Osin for discrete groups, we show that in the case that G is compactly presented relative to H : • if G is compactly generated, then each subgroup H ∈ H is compactly generated; • if each subgroup H ∈ H is compactly presented, then G is compactly presented. The article also introduces an approach to relative hyperbolicity for pairs (G , H) based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, we prove that if G is hyperbolic relative to H then G is compactly presented relative to H. As an application of the results of the article we prove combination results for coherent topological groups with a compact open subgroup, and extend McCammond-Wise perimeter method to this general framework. [ABSTRACT FROM AUTHOR]

We show that groups presented by inverse-closed finite confluent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate geodesic triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of Z) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite confluent length-reducing rewriting system is not plain is in NP. A 'yes' answer to an instance of this problem would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite confluent length-reducing rewriting systems is in PSPACE. • A new geometric characterisation of groups presented by inverse-closed finite confluent length-reducing rewriting systems. • A new algebraic characterisation of groups presented by inverse-closed finite confluent length-reducing rewriting systems. • Deciding if a group presented by inverse-closed finite confluent length-reducing rewriting systems is not plain is in NP. • Deciding isomorphism between plain groups given by inverse-closed finite confluent length-reducing rewriting systems is in PSPACE. [ABSTRACT FROM AUTHOR]

We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem. [ABSTRACT FROM AUTHOR]

We provide the spectral picture of groups of weighted composition operators, induced by the hyperbolic group of automorphisms of the unit disc, acting on holomorphic functions. Some questions about the spectrum of single weighted hyperbolic composition operators are discussed, and results related with them in the literature are completed or partly extended. Also, our results on the weighted hyperbolic group are applied to the spectral study of two families of multiparameter weighted averaging operators, which generalize both Siskakis' operator and the reduced Hilbert matrix operator. [ABSTRACT FROM AUTHOR]

We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeletons of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeletons of finite dimensional \operatorname {CAT}(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group \operatorname {BS}(1,2). Most of these examples satisfy a strong form of the shortcut property. The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions. [ABSTRACT FROM AUTHOR]

We present two explicit combinatorial constructions of finitely summable reduced "Gamma"-elements γr ∈ KK(Cr*(Γ),C) for any word-hyperbolic group (Γ,S) and obtain summability bounds for them in terms of the cardinality of the generating set S U Γ and the hyperbolicity constant of the associated Cayley graph. [ABSTRACT FROM AUTHOR]

We consider groups defined by cyclic presentations where the defining word has length 3 and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known results, this completely classifies the hyperbolic T(6) cyclically presented groups. [ABSTRACT FROM AUTHOR]

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic Λ -metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic Λ -metric spaces, where Λ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case Λ = ℤ n taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ℤ n -metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy : (AMS- 2 0 9) , Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ℤ n -free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ℤ n , Trans. Amer. Math. Soc. 364 (2012) 2847–2882]. [ABSTRACT FROM AUTHOR]

We show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion. [ABSTRACT FROM AUTHOR]

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H (9 , 4) , H (9 , 7). We show that if H (9 , 4) is torsion-free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds. [ABSTRACT FROM AUTHOR]

Groups of F-type were introduced in [B. Fine and G. Rosenberger, Generalizing Algebraic Properties of Fuchsian Groups, London Math. Soc. Lecture Note Ser., 159 (1991) 124{147.] as a natural algebraic generalization of Fuchsian groups. They can be considered as the analogs of cyclically pinched one-relator groups where torsion is allowed. Using the methods In [B. Fine. M. Kreuzer and G. Rosenberger, Faithful Real Representations of Cyclically Pinched One-Relator Groups, Int. J. Group Theory, 3 (2014) 1{8.] we prove that any hyperbolic group of F-type has a faithful representation in PSL(2;R). From this we also obtain that a cyclically pinched one-relator group has a faithful real representation if and only if it is hyperbolic. We further survey the many nice properties of groups of F-type. [ABSTRACT FROM AUTHOR]

Let H be a hyperbolic group, A and B be subgroups of H , and gr (H , A , B) be the growth function of the double cosets A h B , h ∈ H. We prove that the behavior of gr (H , A , B) splits into two different cases. If A and B are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as gr (H , A , B). We can even take A = B. In contrast, for quasiconvex subgroups A and B of infinite index, gr (H , A , B) is exponential. Moreover, there exists a constant λ > 0 , such that gr (H , A , B) (r) > λ f H (r) for all big enough r , where f H (r) is the growth function of the group H. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case. [ABSTRACT FROM AUTHOR]

We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of nonabelian free groups associated with the Patterson-Sullivan measures corresponding to a wide class of invariant metrics on the group are monotonic in the sense introduced by Kuhn and Steger. This in particular includes representations associated with harmonic measures of a wide class of random walks. [ABSTRACT FROM AUTHOR]

A sufficient condition for the hyperbolicity of a group presented in terms of generators and defining relations is considered. The condition is formulated in terms of the negativity of a discrete analog of curvature for the Lyndon-van Kampen diagrams over a presentation of a group and is a generalization of the small cancellation condition. [ABSTRACT FROM AUTHOR]

In [FR 1,2] using faithful complex representations of cyclically pinched andconjugacy pinched one-relator groups we proved that any limit group has afaithful representation in PSL(2;C). Further this representation can be e ec-tively constructed using the JSJ decomposition. In this note we show that anyhyperbolic cyclically pinched one-relator group with maximal amalgamatedsubgroups in each factor has a 2-dimensional faithful real representation.